Measurement of the coefficient of restitution of a golf club

ABSTRACT

A method ( 100 ) and system ( 20 ) for predicting a coefficient of restitution (COR) of a golf club ( 30 ) or golf club head ( 36 ) is disclosed herein. The system ( 20 ) and method ( 100 ) are able to predict the COR in a non-destructive manner for a particular golf ball and impact speed. The system ( 20 ) and method ( 100 ) utilize a vibration sensor ( 24 ) attached to the face ( 34 ) of a golf club ( 30 ). The vibration sensor ( 24 ) is excited with an excitation device ( 26 ), and data is transmitted to an analyzer ( 22 ). A mathematical model of the golf club ( 30 ) is created allowing for the coefficient of restitution to be predicted without destroying the golf club ( 30 ).

CROSS REFERENCES TO RELATED APPLICATIONS

The present application is a continuation-in-part application ofco-pending patent application Ser. No. 09/826544, which was filed onApr. 4, 2001.

STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT

Not Applicable

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates to a system and method for measuring thecoefficient of restitution of a golf club. More specifically, thepresent invention relates to an on-site system and method for measuringthe coefficient of restitution of a golf club without alteration of thegolf club.

2. Description of the Related Art

In 1998, the United States Golf Association (“USGA”) decided to regulatetechnological improvements through a liberal interpretation of Rule 4.1of the Rules of Golf, as set forth by the USGA and the Royals andAncient Club of Saint Andrews. The USGA determined that a golf club headhaving a coefficient of restitution (“COR”) greater than 0.83, on ascale of 0.00 to 1.00, would be non-conforming under the Rules of Golfas a club head having a spring-like effect.

In order to determine the COR of a golf club head, the USGA devised alaboratory test that necessitates the removal of the shaft of a golfclub. The test is conducted at the USGA testing laboratory requiringthat a golf club be submitted to the USGA for conformance. Theun-shafted golf club head is placed on a pedestal without securing theclub head to the pedestal. A PINNACLE® Gold two-piece golf ball is firedat the face of the club head at 160 feet per second. The club head isknocked-off the pedestal, and the COR is measured by the rebound of thegolf ball. A grid is established on the club face using the scorelinesand etched vertical lines, further destroying the club and creatingfurther uncertainties. The procedure is repeated at random sites on thegrid on the face of the golf club until the point with the highest CORis determined from the test. The outbound velocity of the golf ballafter impact with the face is determined using a light gate systems suchas described in U.S. Pat. No. 5,682,230. A more detailed explanation ofthe test is provided at the USGA site.

It is obvious to anyone skilled in the art that such a test isinapplicable to on-course testing, and requires a specific laboratorywith skilled technicians to perform the test. Further, the “cannon test”results in destruction of the club. Yet further, the test is conductedon an unshafted club head, completely ignoring the shaft and grip. Whatis needed is a test that can be performed on course, with consistentrepeatability, and minimal operator error.

BRIEF SUMMARY OF THE INVENTION

The present invention provides a solution to the problems associatedwith testing for COR using the method of the prior art. The presentinvention is able to accomplish this by providing a method and systemthat measures the COR of a golf club in a non-destructive manner, andwithout removal of the shaft.

One aspect of the present invention is a method for predicting thecoefficient of restitution of a golf club. The method includes attachinga vibration sensor to a face of the golf club. Next, the attachedvibration sensor is excited or impacted with an excitation or impactdevice to generate vibrations in the face. Next, the force of impact orthe excitation from the device and the vibrations measured by thevibration sensor are transmitted to an analyzer to generate frequencydomain data for the golf club. Next, the frequency domain data for thegolf club is transformed into a transfer function for the golf club.Then, a golf ball model for a specific golf ball is inputted into thetransfer function along with an impact speed in order to generate apredicted COR for the golf club at a specified impact speed with thespecific golf ball.

Another aspect of the present invention is a system for predicting thecoefficient of restitution of a golf club during impact at a specifiedspeed with a specific golf ball. The system includes an accelerometer,an excitation or impact means, and a calculation means. Theaccelerometer is attached to a point on the face of the golf club. Theaccelerometer has means for measuring the vibration of the face. Themeans for exciting or impacting the face of the golf club in order tovibrate the face has means for measuring the force of excitation orimpact with the face. The calculation means calculates the coefficientof restitution of the golf club from the vibration of the face, theforce of impact or excitation with the face from the impacting orexciting means, an effective mass of the golf club and a mass of a golfball.

Another aspect of the present invention is a system and method forpredicting the coefficient of restitution of a golf club head duringimpact with a golf ball. The system and method is as described above,however, instead of an entire golf club, only the golf club head isutilized for predicting the COR.

Yet another aspect of the present invention is a method for predictingthe coefficient of restitution of a golf club using time domain data.The method includes attaching a vibration sensor to a face of the golfclub. Then, the attached vibration sensor is excited or impacted with anexcitation or impact device to generate vibrations in the face. Next,the force of impact or excitation from the device and the vibrationsmeasured by the vibration sensor are transmitted to an analyzer togenerate time domain data for the golf club. Next, the time domain datafor the golf club is transformed into a state-space representation ofthe golf club. Then, a golf ball model for a specific golf ball isinputted into the state-space representation and an impact speed inorder to generate a predicted COR for the golf club at a specifiedimpact speed with the specific golf ball.

It is a primary object of the present invention to provide a method andapparatus for predicting the COR of a golf club or golf club head.

It is an additional object of the present invention to provide a methodand apparatus for predicting the COR of a golf club in a non-destructivemanner.

It is an additional object of the present invention to provide a methodand apparatus for predicting the COR of a golf club that is portable,and may be performed on-course.

It is an additional object of the present invention to provide a methodand apparatus for predicting the COR of a golf club that requires lesstime than the USGA cannon test.

Having briefly described the present invention, the above and furtherobjects, features and advantages thereof will be recognized by thoseskilled in the pertinent art from the following detailed description ofthe invention when taken in conjunction with the accompanying drawings.

BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWINGS

FIG. 1 is a schematic view of the system of the present invention.

FIG. 2 is a perspective view of a golf club head.

FIG. 3 is a toe end side view of the golf club head of FIG. 2.

FIG. 4 is a bottom view of the golf club head of FIG. 2.

FIG. 5 is a heel end side view of the golf club head of FIG. 2.

FIG. 6 is a side view of a golf club prior to impact with a golf ball.

FIG. 7 is a side view of a golf club during impact with a golf ball.

FIG. 8 is a side view of a golf club after impact with a golf ball.

FIG. 9 is a flow chart of the general method of the present invention.

FIG. 10 is a flow chart of the data acquisition process of the method ofthe present invention.

FIG. 11 is a flow chart of the transformation of the frequency domaindata into a transfer function.

FIG. 12 is schematic diagram of the general model used to identifydynamic linear systems in the frequency domain.

FIG. 13 is a graph of frequency versus magnitude of the fitted curvefrom the frequency domain data.

FIG. 14 is a flow chart of the ball model input into the transferfunction.

FIG. 15 is a chart comparing the predicted values of several golf clubsusing the present invention versus the USGA cannon test values.

FIG. 16 is a chart demonstrating the differences in values between thepresent invention and the USGA cannon test for the golf clubs in thechart of FIG. 15.

DETAILED DESCRIPTION OF THE INVENTION

The present invention is directed at a method and system for measuringthe coefficient of restitution (also referred to herein as “COR”) of agolf club in a non-destructive manner. The COR is generally set forth bythe following equation: $e = \frac{v_{2} - v_{1}}{U_{1} - U_{2}}$

wherein U₁, is the club head velocity prior to impact; U₂ is the golfball velocity prior to impact which is zero; v₁ is the club headvelocity just after separation of the golf ball from the face of theclub head; v₂ is the golf ball velocity just after separation of thegolf ball from the face of the club head; and e is the coefficient ofrestitution between the golf ball and the club face. The values of e arelimited between zero and 1.0 for systems with no energy addition. Thecoefficient of restitution, e, for a material such as a soft clay orputty would be near zero, while for a perfectly elastic material, whereno energy is lost as a result of deformation, the value of e would be1.0.

As shown in FIG. 1, the system of the present invention is generallydesignated 20. The system includes an analyzer 22, a vibration sensor 24and an impact device 26. The vibration sensor 24 and the impact device26 are preferably connected to the analyzer through wires 28 a and 28 b.However, those skilled in the art will recognize that other datatransmission means, such as wireless transmission, could be used withoutdeparting from the scope and spirit of the present invention.

The vibration sensor 24 is attached to a golf club 30 through use of anadhesive 29 such as epoxy, beeswax, or the like. As shown in FIGS. 1-5,the golf club 30 includes a shaft 32, a face 34, a golf club head 36with a body 37. The body 37 of the club head 36 generally includes theface 34, a crown 38 and a sole 40. The club head 36 is partitioned intoa heel section 42 nearest the shaft 32, a toe section 44 opposite theheel section 42, and a rear section 50 opposite the face 34. The face 34has a plurality of scorelines 48 thereon. The club head 36 has a hosel46 for receiving the shaft 32, and the hosel 46 may be internal orexternal.

The club head 36 is typically composed of a stainless steel material ora titanium material. However, those skilled in the art will recognizethat the club head 36 may be composed of other materials such asvitreous metals, ceramics, composites, carbon, carbon fibers and otherfibrous materials. The club head 36 is typically cast or forged, and thethickness of the crown 38, the sale 40 and the face 34 may be constantor varying. Typically, the construction of the face 34 will effect theCOR of the golf club. For example, a high COR. golf club is disclosed inU.S. Pat. No. 6,354,962, filed on Nov. 1, 1999, entitled Golf Club HeadWith Face Composed of a Forged Material, which pertinent parts arehereby incorporated by reference. An example of a low COR golf clubwould be a golf club head composed of a persimmon wood. As shown inFIGS. 6-8, the flexibility of the face 34 allows for a greatercoefficient of restitution. At FIG. 6, the face 34 is immediately priorto striking a golf ball 52. At FIG. 7, the face 34 is engaging the golfball 52, and deformation of the golf ball 52 and face 34 is illustrated.At FIG. 8, the golf ball 52 has just been launched from the face 34.

In a preferred embodiment, the vibration sensor 24 is an accelerometerthat is capable of measuring the vibrations of the face 34 generated byimpact with an impact device 26. An alternative vibration sensor 24 is alaser Doppler vibrometer. The accelerometer may have a titanium cap forprotection during impact. The impact device 26 is preferably a hammerwith a connection to measure the force in volts, and transmit the forceinformation to the analyzer 22 via the wire 28 b. An alternative impactdevice 26 is a fixed striking device that would remove any operatorerror. In an alternative embodiment, the impact device is an excitationdevice that imparts vibrations in the face 34. However, those skilled inthe pertinent art will recognize that other impact or excitation devices26 may be used without departing from the scope and spirit of thepresent invention. The analyzer 22 is preferably a spectrum analyzersuch as an OROS-OR763 spectrum analyzer available from OROS S.A. ofFrance.

The general method is set forth in FIG. 9, which is a flow chart of theoverall method 100. At block 102, the information is acquired from thegolf club 30 by the system 20, as described in greater detail below. Atblock 104, the acquired data is transformed into a transfer functionusing a software such as MATLAB frequency domain system identificationtoolbox, as described in greater detail below. At block 106, thetransfer function is utilized with additional information to create amathematical model of the club. At block 108, either of the followingequations is utilized to generate the COR of the club 30 for a givengolf ball 52 at a predetermined impact speed:

COR=(V′ _(ball) /V _(club))(1+m _(ball) /m _(effective club))−1

COR=(−V′ _(ball) /V _(ball))(1+m _(ball) /m _(effective club))+m_(ball)/m _(effective club)

V_(ball) is the velocity of the golf ball after impact with the club.V′_(ball) is the velocity of the golf ball if it is fired at the golfclub instead of having the golf club swing at a stationary golf ball.V_(club) is the swing speed of the club. m_(ball) is the mass of thegolf ball. m_(effective club) is the effective mass of the golf clubdetermined at zero frequency. The data acquisition process, block 102 ofFIG. 9, is further explained in the flow chart of the FIG. 10. At block110, the club 30 is weighed to obtain the mass of the club 30. At block112, the projection of the center of gravity (“CG”) through the face 34is determined by using known methods of finding the CG, including layingthe club face down on a flat surface. At block 114, the vibration sensor24 is placed on the face 34 at the point where the CG projects throughthe face 34. As mentioned above, the sensor 24 may be mounted usingbeeswax, epoxy, or the like. It is preferred that an adhesive materialwith minimal insulating properties be used for mounting the vibrationsensor 24 in order to allow the vibration sensor 24 to capture as truevibration as possible for the face 34. At block 116, the vibrationsensor 24 is impacted/struck with the impact device 26, and the impactforce, as measured in voltage, is transmitted by the impact device 26 tothe analyzer 22 via the cable 28 b. At block 118, a vibration isgenerated in the face 34 and this vibration is transmitted by the sensor24 to the analyzer 22 via the cable 28 a. At block 120, the analyzer 22transforms the time domain data into a frequency domain equivalent. Theoutput/input ratio (acceleration as indicated by vibration of the face34/the impact force of the hammer 26) is generated by the analyzer 22 ina frequency domain. The frequency domain represents how the system 20reacts at the location of the vibration sensor 24 to a unit impulseinput into the system 20 at the location of the input force by theimpact device 26. This impulse response indicates how the club 30responds to a predetermined force such as impact with a golf ball 52.Those skilled in the art will recognize that time domain data could besubstituted for the frequency domain data. At block 122, a new point maybe chosen on the face 34 for placement of the vibration sensor 24thereto. Then, at block 116, the sensor 24 is impacted with the impactdevice 26 and the cycle is repeated. The cycle may be repeated amultitude of times in order to obtain a grid of the face 34 showing theimpulse response for different points on the face 34. Eventually, thisgrid information could be utilized to obtain the point on the face 34with the highest COR.

The transformation of the frequency domain data into a transferfunction, block 104 of FIG. 9, is further explained in the flow chart ofthe FIG. 11. This transformation is preferably performed using thesystem identification software. System identification is a means fordetermining a model of a physical system. A model that is used foridentifying models for dynamic physical systems is an ordinarydifferential equation or difference equation with constant coefficients.Linear dynamic systems can be described in two regimes: frequency andtime. The frequency domain is preferred due to the proliferation of thedigital computer and the fast Fourier transform. The general model usedto identify dynamic linear systems in the frequency domain is set inFIG. 12. The transfer function H(Ω) represents the club 30 whereΩ=s=jω=j2πf. “f” is the frequency and “j” is an imaginary number squareroot of negative one. “ω”is the angular frequency. Xm and Ym representthe measured input and output complex amplitudes respectively. Thesevalues are the combination of noise (Nx and Ny) superimposed upon thetheoretical “true” input and output amplitudes X and Y. The transferfunction, in its rational fraction form, is as follows:${H(\Omega)} = {^{{- {j\omega}}\quad T_{d}}\frac{{b_{0}\Omega^{0}} + {b_{1}\Omega^{1}} + \ldots \quad + {b_{nn}\Omega^{nn}}}{{a_{0}\Omega^{0}} + {a_{1}\Omega^{1}} + \ldots \quad + {a_{nd}\Omega^{nd}}}}$

Advanced curve-fitting and cost function optimization is utilized toderive rational fraction polynomial numerator and denominatorcoefficients (b_(n) and a_(n)) in the equation above. These coefficientsare then used to form a theoretical transfer function that bestrepresents the true model Y/X. This is then used to find the best fit toa measured transfer function that represents the physical system.

Referring again to FIG. 11, at block 130, a graph of frequency versusmagnitude of transfer accelerance (1/kg) is generated from the frequencydomain data, which is shown in FIG. 13. At block 132, the phase isanalyzed to determine when the face 34 is traveling inward and outward.At block 134, a frequency band that is the best mathematicalrepresentation of the club 30 and minimizes noise is selected from thegraph. The graph is analyzed to find the first and largest magnitudepeak 131, which is the first vibration of the face 34. The graph is alsoanalyzed to find the first anti-resonance 133, which is similar to anenergy sink. The first inflection point 135 is also determined from thegraph. Due to vibrations at higher frequencies which correspond to thecrown 38, the sole, the face 34 or any combination thereof, the bestmathematical representation of the club 30 is the information from 500Hertz (“Hz”) to the first inflection point 135. At block 136, the systemidentification calculation is performed by the software. At block 138,the transfer function is generated from the fitted information from thegraph.

In FIG. 12, the effective mass of the club 30 can be obtained at zerofrequency since according to Newton's Second Law,Force=mass×acceleration, (F=ma) and 1/m=a/F. The effective mass iscompared to the measured mass of the club.

It should be noted that the COR of a golf club 30 is dependent on thegolf ball and speed of impact, and thus, the COR of a golf club willvary if the golf ball is changed or if the speed is changed. Forexample, if the COR is measured for a golf club 30 using a CALLAWAYGOLF® RULE 35® SOFTFEEL™ golf ball at a speed of 110 miles per hour(“mph”), the COR will be different than that measured using a CALLAWAYGOLF® CB1™ golf ball at a speed of 110 mph. Additionally, if the COR ismeasured for a golf club 30 using a CALLAWAY GOLF® RULE 35® SOFTFEEL™golf ball at a speed of 110 mph, the COR will be different than thatmeasured using a CALLAWAY GOLF® RULE 35® SOFTFEEL™ golf ball at a speedof 85 mph.

In order for the method and system 20 of the present invention tooperate in a non-destructive manner, at block 106 of FIG. 9, a golf ballmodel must be used with the transfer function in order to have a meansfor generating the COR for a golf club 30. Just like the transferfunction is unique to a specific golf club 30, the golf ball model isunique to a specific golf ball. Thus, the golf ball model for the aCALLAWAY GOLF® RULE 35® SOFTFEEL™ golf ball (a three-piece solid golfball with a very thin thermoset polyurethane cover) is different than aCALLAWAY GOLF® CB1™ golf ball (a two-piece golf ball with an ionomerblend cover). The golf ball model is obtained by recording: the inboundspeed and outbound speed for a specific golf ball fired at a zero degreeloft striking plate; the contact duration; the contact force; and theCOR of the golf ball. This data is obtained for different speeds and anon-linear ball model is created such as disclosed in Chapter 61 ofAlastair Cochran's Science and Golf III, Proceedings of the 1998 WorldScientific Congress of Golf, Human Kinetics 1999.

As shown in the flow chart of FIG. 14, at block 150, the force output ofthe golf ball model is inputted to the transfer function. At block 152,the club speed with the output from block 150 generates a contact pointvelocity. At block 154, the contact point velocity is integrated to givethe contact point displacement, which is a representation of the bendingof the face 34 about the golf ball 52.

This information is used in block 108 of FIG. 9 to input into thepreviously mentioned equations to obtain the COR:

COR=(V′ _(ball) /V _(club))(1+m _(ball) /m _(effective club))−1

COR=(−V′ _(ball) /V _(ball))(1+m _(ball) /m _(effective club))+m_(ball)/m _(effective club)

For example, a CALLAWAY GOLF® HAWK EYE® nine degree driver (club headonly) was tested using the present invention. The driver had a measuredmass of 195.9 grams and an effective mass of 193.9 grams. Using acontact point velocity of 21.9 feet per second, and a V_(out)/V_(in)ratio of 0.4485, the COR was predicted to be 0.7905.

FIG. 15 illustrates a chart of the COR values generated from the methodand system of the present invention as compared to the COR valuesgenerated from the USGA cannon test as previously described and as setforth on the USGA web site. The nine club heads are composed oftitanium, titanium alloy, stainless steel, or in the case of theCALLAWAY® CLASSIC® golf club, persimmon wood. FIG. 16 illustrates thedifference in values between the method and system of the presentinvention and the USGA cannon test for the nine clubs. The actualnumbers are provided in Table One below. As mentioned above, the presentinvention may be utilized for clubs or for club heads. In order toprovide a more accurate comparison to the USGA test, the information inFIGS. 15 and 16, and in Table One is for club heads. As is apparent fromFIG. 16, the predicted COR from the present invention is within ±0.008of the USGA values.

TABLE ONE Ball Head Predicted USGA Club Head mass mass COR COR CallawayClassic 0.0456 0.19813 0.775 0.778 Callaway Steelhead Plus 9° 0.04560.20106 0.795 0.793 Taylor Made Firesole 9.5° 0.0456 0.20377 0.805 0.803PING Ti ISI 8.5° 0.0456 0.19974 0.809 0.811 Taylor Made Firesole 32010.5° 0.0456 0.18855 0.820 0.817 Mizuno Pro 300S 10° 0.0456 0.199980.820 0.821 Katana Sword 300Ti 10° 0.0456 0.1924  0.832 0.833Bridgestone Break the Mode 10° 0.0456 0.19672 0.840 0.843 Daiwa G-3Hyper Titan 10.5° 0.0456 0.18781 0.852 0.855

From the foregoing it is believed that those skilled in the pertinentart will recognize the meritorious advancement of this invention andwill readily understand that while the present invention has beendescribed in association with a preferred embodiment thereof, and otherembodiments illustrated in the accompanying drawings, numerous changes,modifications and substitutions of equivalents may be made thereinwithout departing from the spirit and scope of this invention which isintended to be unlimited by the foregoing except as may appear in thefollowing appended claims. Therefore, the embodiments of the inventionin which an exclusive property or privilege is claimed are defined inthe following appended claims.

We claim as our invention:
 1. A method for predicting a coefficient ofrestitution (COR) of a golf club, the method comprising: attaching avibration sensor to a face of the golf club; impacting the attachedvibration sensor with an excitation device to generate vibrations in theface; transmitting a force of excitation from the excitation device andthe vibrations measured by the vibration sensor to an analyzer togenerate frequency domain data for the golf club; generating a graph offrequency versus magnitude of transfer accelerance from the frequencydomain data; analyzing the phase of the graph of frequency to determinewhen the face of the golf club is traveling inward and outward fordetermining a contact point velocity; selecting a frequency band tominimize noise, the frequency band ranging from 500 Hertz to a firstinflection point of the graph; calculating a transfer function from thefrequency band of the graph; and inputting a golf ball model for aspecific golf ball and an impact speed into the transfer function thatcontains an acceleration magnitude and direction of the face of the golfclub in order to generate a predicted COR for the golf club at theimpact speed with the specific golf ball, the golf ball model comprisinga contact duration of the specific ball, a contact force of the specificball, a COR of the specific ball and a mass of the specific ball.
 2. Themethod according to claim 1 wherein the impact speed is a club speed. 3.The method according to claim 1 wherein the impact speed is a ballspeed.
 4. The method according to claim 1 wherein the vibration sensoris an accelerometer with the capability of transmitting data to theanalyzer.
 5. The method according to claim 1 wherein the excitationdevice is an impact device with the capability of generating the forceof excitation and transmitting the force of excitation to the analyzer.6. The method according to claim 1 further comprising inputting aneffective mass of the golf club into the transfer function, theeffective mass calculated from the graph of frequency at 0 Hertz.
 7. Themethod according to claim 1 wherein a system identification is used togenerate the transfer function.
 8. A method for predicting a coefficientof restitution (COR) of a golf club head, the method comprising:attaching an accelerometer to a face of the golf club head; impactingthe attached accelerometer with an impact device to generate vibrationsin the face; transmitting a force of impact from the impact device andthe vibrations measured by the accelerometer to an analyzer to generatefrequency domain data for the golf club head; generating a graph offrequency versus magnitude of transfer accelerance from the frequencydomain data; analyzing the phase of the graph of frequency to determinewhen the face of the golf club is traveling inward and outward fordetermining a contact point velocity; analyzing the graph of frequencyto determine a first anti-resonance; selecting a frequency band tominimize noise, the frequency band ranging from 500 Hertz to a firstinflection point of the graph; calculating a transfer function from thefrequency band of the graph; and inputting a golf ball model for aspecific golf ball and an impact speed into the transfer function thatcontains an acceleration magnitude and direction of the face of the golfclub in order to generate a predicted COR for the golf club head at theimpact speed with the specific golf ball, the golf ball model comprisinga contact duration of the specific ball, a contact force of the specificball, a COR of the specific ball and a mass of the specific ball.
 9. Themethod according to claim 8 further comprising inputting an effectivemass of the golf club head into the transfer function, the effectivemass calculated from the graph of frequency at 0 Hertz.